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Estimation of Tsallis entropy for exponentially distributed several populations

Naveen Kumar, Ambesh Dixit, Vivek Vijay

TL;DR

The paper addresses estimating the Tsallis entropy for a finite set of exponential populations with a common scale and distinct shifts, focusing on the function Theta(sigma)=1/sigma^(k(q-1)). It derives the BAEE under a bowl-shaped quadratic loss and proves its inadmissibility via Stein-type improvements, then develops smooth Brewster-Zidek estimators and a Bayes estimator with an inverse-gamma prior. Through theoretical results and simulations, it shows the Brewster-Zidek estimator often outperforms BAEE and Stein-type in finite samples, with improvements diminishing as the sample size increases. The findings provide practical, parametric procedures for entropy estimation in multi-population exponential models and suggest applicability to other bowl-shaped loss functions.

Abstract

We study the estimation of Tsallis entropy of a finite number of independent populations, each following an exponential distribution with the same scale parameter and distinct location parameters for $q>0$. We derive a Stein-type improved estimate, establishing the inadmissibility of the best affine equivariant estimate of the parameter function. A class of smooth estimates utilizing the Brewster technique is obtained, resulting in a significant improvement in the risk value. We computed the Brewster-Zidek estimates for both one and two populations, to illustrate the comparison with best affine equivariant and Stein-type estimates. We further derive that the Bayesian estimate, employing an inverse gamma prior, which takes the best affine equivariant estimate as a particular case. We provide a numerical illustration utilizing simulated samples for a single population. The purpose is to demonstrate the impact of sample size, location parameter, and entropic index on the estimates.

Estimation of Tsallis entropy for exponentially distributed several populations

TL;DR

The paper addresses estimating the Tsallis entropy for a finite set of exponential populations with a common scale and distinct shifts, focusing on the function Theta(sigma)=1/sigma^(k(q-1)). It derives the BAEE under a bowl-shaped quadratic loss and proves its inadmissibility via Stein-type improvements, then develops smooth Brewster-Zidek estimators and a Bayes estimator with an inverse-gamma prior. Through theoretical results and simulations, it shows the Brewster-Zidek estimator often outperforms BAEE and Stein-type in finite samples, with improvements diminishing as the sample size increases. The findings provide practical, parametric procedures for entropy estimation in multi-population exponential models and suggest applicability to other bowl-shaped loss functions.

Abstract

We study the estimation of Tsallis entropy of a finite number of independent populations, each following an exponential distribution with the same scale parameter and distinct location parameters for . We derive a Stein-type improved estimate, establishing the inadmissibility of the best affine equivariant estimate of the parameter function. A class of smooth estimates utilizing the Brewster technique is obtained, resulting in a significant improvement in the risk value. We computed the Brewster-Zidek estimates for both one and two populations, to illustrate the comparison with best affine equivariant and Stein-type estimates. We further derive that the Bayesian estimate, employing an inverse gamma prior, which takes the best affine equivariant estimate as a particular case. We provide a numerical illustration utilizing simulated samples for a single population. The purpose is to demonstrate the impact of sample size, location parameter, and entropic index on the estimates.
Paper Structure (7 sections, 9 theorems, 79 equations, 4 figures, 2 tables)

This paper contains 7 sections, 9 theorems, 79 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. BAEE for the Tsallis entropy of multiple exponential populations
  3. A Stein-type Improved Equivariant Estimate
  4. A Smooth Improved Equivariant Estimate
  5. Bayes Estimator
  6. Numerical Comparisons
  7. Concluding Remarks

Key Result

Lemma 2.1

Given $k$ independent exponential distributions as in equation(twoparaExponentialdistr), the joint Tsallis entropy is $S_q(P_1,P_2,...,P_k)=\frac{1}{q-1}\left(1-{\left[1+(1-q)S_q(P_i)\right]}^k\right)$.

Figures (4)

  • Figure 1: PRI values of Brewster-Zidek estimate for a simulated sample of size $n=4, \sigma=1$, $u=0,0.1,0.2,0.3,0.4,0.5$ and $q>0.$
  • Figure 2: Risk value plot of BAEE, Stein-type and Brewster-Zidek estimator for $\sigma=1, u=0.1, q=0.1$ and $50$ samples of sizes $4$
  • Figure 3: Risk value plot of BAEE, Stein-type and Brewster-Zidek estimator for $\sigma=1, u=0.1, q=0.1$ and $50$ samples of sizes $8$
  • Figure 4: PRI values plot of Stein-type and Brewster-Zidek estimator for $\sigma=1, u=0.1, q=0.1$ and sample size vary from $2$ to $50$

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Lemma 3.1
  • proof
  • ...and 13 more